Robert Hansen says: >There is no such thing as a "common sense view of continuity".
Perhaps not for you - only for the rest.
Robert Hansen says: >What you are trying to say is that a function whose graph can be traced without lifting the pencil is a common sense "interpretation" of continuity. But by itself it is not an interpretation of anything. It is only a damn good interpretation to you and I because we already understand continuity.
Nonsense, mathematicians tacitly understood continuity long before epsilons and deltas. Zeno came up with his famous paradoxes precisely because he saw the trap laying in wait for those who tried to precisely analyze the intuitive concept that all or at least most humans are capable of imagining. The pencil illustration almost perfectly captures that intuition - no gaps.